The quadrupole interaction of positronium in noncubic defectless crystals has been investigated theoretically. It is shown to lead to the weak anisotropy of positronium magnetic quenching. The temperature dependence of the quadrupole coupling constant and the estimations of the magnitude of the anisotropy are obtained for delocalized positronium in crystalline quartz. The significant magnitude of the anisotropy is predicted for "anomalous" positronium in oriented crystalline polymeric structures.